296 ANALYTICAL MECHANICS 3. A gun can project a shot to a height of ~, where R is the radius of the earth. Taking the variation of the gravitational force with altitude, show that the gun can command -5 of the earth's surface. 4. A particle is projected into a smooth horizontal circular groove. The particle is attracted towards a point in the radius which joins the position of projection with the center, with a force equal to ^ • Show that in order that the particle may be able to make complete revolutions the initial velocity must not be less than~~~77> where a is the radius of the groove and b the distance of the center of force from the center of the groove. 5. A comet describing a parabolic orbit about the sun collides with a body of equal mass at rest. Show that the center of mass of the two describes a circle about the sun as center. 6. Prove that the least velocity with which a bod}'' must be projected from the north pole so as to hit the surface of the earth at the equator is about 4-J miles per second, and that the angle of elevation is 22°.5. 7. A particle moves in the common Held of two fixed centers of force of equal intensity. The particle is attracted towards one of the centers with a force which varies as its distance from that center, and repelled from the other center according to the same law. Show that the orbit is a parabola. 8. A particle moves in a field in which the force is repulsive and varies inversely as the square of the distance from the center of force. Show that the orbit is a hyperbola. 9. In the preceding problem show that the radius vector sweeps over equal areas in equal intervals of time.